Problem 15 - Algebra 2

Question

A map of Alaska is scaled so that 3 inches represents 93 miles. How far apart are Anchorage and Fairbanks if they are 11.6 inches apart on the map?

Step-by-Step Solution

Answer

Anchorage and Fairbanks are 360.8 miles apart.

Step 1: Understand the Scale

The scale of the map indicates that 3 inches on the map corresponds to 93 miles in reality. This provides us with a ratio that we can use to solve the problem.

Step 2: Set Up the Proportion Equation

Using the scale, set up a proportion to find out how many miles the 11.6 inches represents. The proportion can be written as \( \frac{3}{93} = \frac{11.6}{x} \) where \( x \) is the number of miles that corresponds to 11.6 inches.

Step 3: Solve for x

To solve for \( x \), cross-multiply the equation: \[ 3 \times x = 93 \times 11.6 \] Then divide both sides by 3:\[ x = \frac{93 \times 11.6}{3} \] Calculate the right side to find the value of \( x \).

Step 4: Calculate the Distance

Now, compute the expression from Step 3:\[ x = \frac{93 \times 11.6}{3} = 360.8 \] Thus, the distance between Anchorage and Fairbanks is 360.8 miles.

Key Concepts

Proportion Equation

In the context of map scales, a proportion equation forms the foundation for solving distance problems. By understanding that map distance and real distance relate proportionally, you can easily set up such an equation.
To produce a proportion equation, you start by identifying two pairs of equivalent ratios. In this case, we know that 3 inches on the map equates to 93 miles in reality. If you express these two values as fractions, the equation becomes \( \frac{3}{93} = \frac{11.6}{x} \), where \( x \) will be the unknown real distance.
Using proportion equations is essential when dealing with map scales because they maintain consistent relationships, ensuring accuracy in distance conversion. By setting up a valid proportion, you can effectively link two different measurements and solve for unknown variables.

Cross-Multiplication

Cross-multiplication is a crucial technique for solving proportion equations. It allows you to eliminate fractions and solve for a variable quickly and efficiently.
When you have a proportion like \( \frac{3}{93} = \frac{11.6}{x} \), cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other.
So, you would calculate \( 3 \times x = 93 \times 11.6 \). This process simplifies the equation by removing fractions, allowing the equation to be easier to manage and understand.
Cross-multiplication is particularly useful in algebraic problem-solving because it directly tackles the problem of fractions, turning a complex-looking equation into a simpler one, where you can directly solve for the unknown variable.

Distance Calculation

Distance calculation in map scale problems requires converting map measurements into real-world distances. After setting up and rearranging your proportion equation using cross-multiplication, you will solve for the distance.
In the scenario given, the equation \( 3 \times x = 93 \times 11.6 \) led to simplifying \( x = \frac{93 \times 11.6}{3} \). Calculating this expression gives \( x = 360.8 \), revealing the real-world distance.
Understanding how to calculate distances isn't just about plugging numbers into formulas; it also involves comprehending the relationship between the scale and actual geography. Once you interpret this relationship correctly, the calculation follows naturally.

• Ensure that you always double-check your final unit conversion.
• Verify whether the distance is presented in the same unit as the map scale.

With these checks, your calculated distances will accurately represent real-world measurements.

Algebraic Problem Solving

Algebraic problem-solving skills are the backbone of handling map scale issues or any mathematical challenges. By using algebra, you establish a logical framework to handle unknowns and variables.
In the map scale example, algebra enables you to set the equation \( \frac{3}{93} = \frac{11.6}{x} \) and systematically solve for \( x \). This approach harnesses basic algebraic principles like balancing equations and manipulating terms to isolate variables.
Effective algebraic problem-solving lays the groundwork for resolving a wide range of practical problems, whether you're converting units or balancing more complex mathematical inequalities.

• Familiarize yourself with fundamental algebraic concepts, such as variables and operations.
• Practice simplifying expressions, setting up equations, and analyzing results for consistent accuracy.

These skills ensure you can leverage algebra not just for map scales but for any proportion-based problem you encounter.